diff options
Diffstat (limited to '')
| -rw-r--r-- | hott/Nat.thy | 14 |
1 files changed, 7 insertions, 7 deletions
diff --git a/hott/Nat.thy b/hott/Nat.thy index c36154e..e8657a4 100644 --- a/hott/Nat.thy +++ b/hott/Nat.thy @@ -18,24 +18,24 @@ where NatE: "\<lbrakk> n: Nat; c\<^sub>0: C 0; - \<And>k c. \<lbrakk>k: Nat; c: C k\<rbrakk> \<Longrightarrow> f k c: C (suc k); + \<And>k rec. \<lbrakk>k: Nat; rec: C k\<rbrakk> \<Longrightarrow> f k rec: C (suc k); \<And>n. n: Nat \<Longrightarrow> C n: U i - \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) n: C n" and + \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) n: C n" and Nat_comp_zero: "\<lbrakk> c\<^sub>0: C 0; - \<And>k c. \<lbrakk>k: Nat; c: C k\<rbrakk> \<Longrightarrow> f k c: C (suc k); + \<And>k rec. \<lbrakk>k: Nat; rec: C k\<rbrakk> \<Longrightarrow> f k rec: C (suc k); \<And>n. n: Nat \<Longrightarrow> C n: U i - \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) 0 \<equiv> c\<^sub>0" and + \<rbrakk> \<Longrightarrow> NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) 0 \<equiv> c\<^sub>0" and Nat_comp_suc: "\<lbrakk> n: Nat; c\<^sub>0: C 0; - \<And>k c. \<lbrakk>k: Nat; c: C k\<rbrakk> \<Longrightarrow> f k c: C (suc k); + \<And>k rec. \<lbrakk>k: Nat; rec: C k\<rbrakk> \<Longrightarrow> f k rec: C (suc k); \<And>n. n: Nat \<Longrightarrow> C n: U i \<rbrakk> \<Longrightarrow> - NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) (suc n) \<equiv> - f n (NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k c. f k c) n)" + NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) (suc n) \<equiv> + f n (NatInd (\<lambda>n. C n) c\<^sub>0 (\<lambda>k rec. f k rec) n)" lemmas [intros] = NatF Nat_zero Nat_suc and |